Structure and dynamics of liquid helium systems and their interaction with atomic dopants and free electrons

Structure and dynamics of liquid helium systems and their interaction with atomic

dopants and free electrons

David Mateo Valderrama

Aquesta tesi doctoral està subjecta a la llicència Reconeixement- CompartIgual 3.0. Espanya de Creative Commons.

Esta tesis doctoral está sujeta a la licencia Reconocimiento - CompartirIgual 3.0. España de Creative Commons.

This doctoral thesis is licensed under the Creative Commons Attribution-ShareAlike 3.0. Spain License.

dopants and free electrons

by

David Mateo Valderrama

dopants and free electrons

by

David Mateo Valderrama

Supervisor: Dr. Manuel Barranco G´omez Tutor: Dr. Mart´ı Pi Pericay

Thesis submitted for the degree of Doctor of Philosophy under the Physics PhD program at the University of Barcelona.

March 2013

Tesis presentada para obtener el grado de Doctor en el pro- grama de doctorado de F´ısica de la Universidad de Barcelona.

Marzo de 2013

Contexto

El helio es la ´ unica sustancia que permanece l´ıquida a temperatura cero. Esta curiosa caracter´ıstica es una clara manifestaci´ on macrosc´ opica de la naturaleza cu´ antica de la materia. El helio es el segundo elemento m´ as ligero y su estructura de capa cerrada es responsable de la d´ ebil atracci´ on entre ´ atomos de He. Estas dos propiedades son la causa de que el helio tenga una energ´ıa de punto cero suficientemente grande como para evitar la solidificaci´ on, incluso a temperatura estrictamente cero. La energ´ıa del punto cero –tambi´ en conocida como movimiento del punto cero– es la energ´ıa asociada a la localizaci´ on de la distribuci´ on de probabilidad de un sistema. La existencia de esta energ´ıa es una predicci´ on de la mec´ anica cu´ antica sin equivalente cl´ asico que desempe˜ na el papel de energ´ıa cin´ etica incluso cuando no hay “movimiento” en el sentido cl´ asico. La naturaleza cu´ antica del He se manifiesta no solo impidiendo su solidificaci´ on sino tambi´ en causando una transici´ on de segundo orden cuando la temperatura se reduce por debajo de los 2.18 grados Kelvin de un estado l´ıquido normal (He I) a un estado superfluido (He II); un nuevo estado de la materia en el que la ausencia de viscosidad permite al helio fluir libre- mente (sin disipaci´ on) cuando se desplaza a velocidades por debajo de cierto valor l´ımite.[1]

El helio aparece tanto en forma de l´ıquido homog´ eneo como en forma de gotas. Las gotas pueden estar formadas por cualquiera de los dos is´ otopos estables del helio,

He y

4

He, o de una mezcla de ambos. A bajas temperaturas, una gota mixta de

He–

He es un agregado estructurado con

He en su n´ ucleo y

He en su capa externa. Una propiedad interesante de estas gotas es que pueden ser dopadas con impurezas de forma relativa- mente f´ acil. El estudio espectrosc´ opico de estas impurezas permite analizar el entorno de helio que las rodea. [2–7] Dependiendo de la impureza, ´ esta se situar´ a bien en el centro de la gota, en la superficie, o en la interficie

He–

He.[8] La estructura de gotas de

He,

4

He y, en menor medida,

He–

He alrededor de impurezas at´ omicas o electrones libres ha sido estudiada tanto a nivel te´ orico como experimental.

La din´ amica de estos sistemas se ha resistido a una exploraci´ on te´ orica durante muchos a˜ nos. Uno podr´ıa argumentar que, dado los pocos experimentos llevados a cabo sobre la respuesta del helio en la escala del picosegundo, una descripci´ on precisa de la din´ amica es una cuesti´ on puramente acad´ emica de poca relevancia para la comunidad

i

experimental. Sin embargo, diversos experimentos como por ejemplo la observaci´ on de la desaparici´ on de burbujas electr´ onicas excitadas[9] por el Prof. Maris o la medici´ on de la velocidad de desorci´ on de ´ atomos de Ag dentro de gotas[10] del Prof. Drabbels muestran que una descripci´ on te´ orica completa y precisa del proceso din´ amico que subyace a estas experiencias es fundamental para comprender los resultados.

Resultados

En esta tesis se presenta una colecci´ on de cuatro art´ıculos publicados y un manuscrito a´ un no publicado, todos ellos en el campo de la f´ısica de bajas temperaturas y fluidos cu´ anticos. Cada uno de ellos reporta un paso adelante en la descripci´ on te´ orica de los sistemas de helio por medio de la teor´ıa del funcional de la densidad.

Los primeros dos art´ıculos [11, 12] est´ an clasificados como “estructura” ya que tratan cuestiones relacionadas con la descripci´ on del estado fundamental de complejos de helio dopados con impurezas at´ omicas. En ellos hemos calculado la estructura y determinado su efecto sobre el espectro de absorci´ on dipolar del Na en agregados de

He–

He y del Mg en el l´ıquido homog´ eneo e isot´ opicamente mezclado. Para el caso de Na en gotas se ha encontrado que, a pesar de necesitar una gran cantidad de

He para que la capa exterior de la gota sature, el espectro de la impureza es muy insensible a la composici´ on isot´ opica y r´ apidamente satura al valor que toma en las gotas de

He puras. Para Mg en el l´ıquido mezclado, la presencia de

He induce cambios en el espectro mucho m´ as peque˜ nos que su anchura caracter´ıstica, por lo que se ha encontrado que el efecto general de la composici´ on isot´ opica de la mezcla en la espectroscop´ıa es m´ınima.

Hemos explorado tambi´ en los l´ımites del funcional de la densidad para un n´ umero peque˜ no de ´ atomos de helio interactuando con una mol´ ecula lineal de sulfuro de carbonilo (OCS). Para ello hemos implementado un esquema de Kohn-Sham para el

He y hemos obtenido la estructura de agregados OCS@

He

para un n´ umero de ´ atomos N hasta 40.

Hemos comparado los resultados de los agregados de

He con el mismo n´ umero de ´ atomos y hemos encontrado que la alta anisotrop´ıa de la mol´ ecula de OCS magnifica los efectos de las diferentes estad´ısticas de cada is´ otopo. Nuestra estimaci´ on de los momentos de inercia de estos agregados es consistente con la interpretaci´ on de los datos experimentales[13]

que sugieren una estructura de 11 ´ atomos de helio rotando solidariamente con la mol´ ecula de OCS.

Los siguientes tres trabajos [10, 14, 15], clasificados como “din´ amica”, describen la evoluci´ on temporal de ciertos procesos de inter´ es experimental en los sistemas de helio.

Mientras que las publicaciones sobre la estructura completan una l´ınea de trabajo bien

establecida, las de esta secci´ on abren un nuevo frente de exploraci´ on te´ orica sobre los

procesos din´ amicos con resoluci´ on de picosegundos. En ellos se presenta un procedimien-

to eficiente para describirlos cuantitativamente mediante una teor´ıa del funcional de

la densidad dependiente del tiempo (TDDFT, por sus siglas en ingl´ es) para el helio,

acoplado a la din´ amica adecuada para la impureza. Cu´ al es la din´ amica “adecuada”

utilizar una descripci´ on puramente mecanocu´ antica del electr´ on en una aproximaci´ on adiab´ atica, mientras que para la burbuja 2P la aproximaci´ on adiab´ atica no es aplicable y se deben acoplar las evoluciones en tiempo real del helio y del electr´ on. Para una impureza masiva como la Ag una descripci´ on cl´ asica de su movimiento es suficiente, pero el efecto de spin-´ orbita es lo suficientemente importante como para que el estado electr´ oni- co del ´ atomo deba tenerse en cuenta en la din´ amica como un grado de libertad cuantizado.

En el caso de las burbujas electr´ onicas, hemos relacionado la desaparici´ on de las burbujas 1P a altas presiones con la existencia de un camino de relajaci´ on no-radiativo que causa la rotura de la burbuja en dos mitades casi esf´ ericas tras haber transcurrido unos 20 picosegundos desde su excitaci´ on. Hemos sido capaces de establecer esta relaci´ on gracias a que nuestro c´ alculo predice la “fisi´ on” de la burbuja solo para presiones superiores a 1 bar, que es el mismo umbral observado experimentalmente para la desaparici´ on de las burbujas 1P. [9] Teniendo en cuenta que aumentar la presi´ on en 1 bar aumenta la densidad de saturaci´ on del l´ıquido en poco m´ as de un 1 %, la exactitud de este resultado indica que TDDFT contiene la f´ısica relevante para describir este tipo de procesos y tiene capacidad de predicci´ on cuantitativa. Tambi´ en hemos encontrado un marcado cambio en el comportamiento del espectro de absorci´ on con resoluci´ on temporal de la burbuja 1P dependiendo de si ´ esta fisiona o no, es decir, dependiendo de la presi´ on del l´ıquido.

La observaci´ on experimental de este cambio y su determinaci´ on podr´ıa completar la informaci´ on obtenida en los experimentos de cavitaci´ on y fotoconductividad.

En el caso de la desorci´ on de un ´ atomo de plata tras su fotoexcitaci´ on en el interior

de una gota de He, nuestros c´ alculos din´ amicos predicen un rango de velocidades para

la impureza consistente con la distribuci´ on de velocidades experimental. Esta velocidad

es el resultado de cu´ anta energ´ıa transfiere la impureza a la gota, lo cual depende de

los modos de excitaci´ on disponibles para dicha transferencia. Nuestra descripci´ on del

l´ıquido solo permite excitaciones colectivas tales como ondas de densidad u oscilaciones de

superficie, por lo que la compatibilidad de nuestros resultados con los datos experimentales

se puede tomar como una evidencia indirecta de la superfluidez de las nanogotas de

helio. Adicionalmente hemos descartado la nucleaci´ on v´ ortices como un posible canal

de transferencia de energ´ıa en gotas nanosc´ opicas al no haberlos generado en nuestros

c´ alculos.

Resumen i

1 Introduction 1

1.1 Theoretical description of helium systems . . . . 2 1.1.1 Density Functional Theory . . . . 3

2 Structure 5

2.1 Absorption spectrum of atomic impurities in isotopic mixtures of liquid helium . . . . 5 2.2 A density functional study of the structure of small OCS@

He

clusters 14

3 Dynamics 23

3.1 Evolution of the excited electron bubble in liquid

He and the appearance of fission-like processes . . . . 23 3.2 Excited electron-bubble states in superfluid

He: A time-dependent density

functional approach . . . . 34 3.3 Translational dynamics of photoexcited atoms in

He nanodroplets: The

case of silver . . . . 44

4 Summary and Conclusions 59

4.1 Outlook . . . . 61

A List of publications 63

Bibliography 65

v

Introduction

Helium first revealed itself in 1868 as a bright yellow line coming from the Sun. As the unworldly substance it seemed to be, it was named after the Greek word for Sun, helios.

Fourteen years later its presence on Earth was unraveled by analyzing the spectral lines of the lava on Mount Vesuvius. Before the turn of the century, helium would be captured and isolated for study in several laboratories around Europe. In 1908, Heike Kamerlingh Onnes achieved the liquefaction of helium by cooling it below four degrees Kelvin, but helium would always defy solidification under normal pressure.[16] This seemingly inno- cent realization marks the start for an exhaustive research which would reveal a new state of matter and surprising new phenomenology in the field of low temperature physics.

Helium is the only substance which remains liquid at zero temperature. This striking feature is a clear and indisputable macroscopic manifestation of the quantum nature of matter. Helium is the second lightest element and its closed shell structure is responsible for the extremely small attraction between He atoms. These two properties cause the helium to have a zero point energy large enough to keep it from solidifying even at strictly zero temperature. The zero point energy –also known as zero point motion– is the energy associated with the localization of the probability distribution of a system on a finite region of space. The existence of this energy is a prediction of Quantum Mechanics with no classical equivalent, and plays the role of a kinetic energy present even when there is no “motion” in the classical sense. The quantum nature of He does not only prevents it from solidifying but also causes a second order phase transition when the temperature is reduced below 2.18 Kelvin from normal liquid state (He I) to a superfluid state (He II);

a new state of matter in which the absence of viscosity allows it to flow freely (with no dissipation) at velocities below a certain limit.[1]

The superfluid state is a highly correlated state where particle excitations are sup- pressed in favor of delocalized collective excitations. It is associated with Bose-Einstein condensation, a manifestation of the bosonic nature of helium’s most abundant isotope,

4

He. Helium has a second stable isotope,

He, which has spin 1/2 and therefore follows fermionic statistics. The different statistics have a critical effect on the macroscopic

1

properties of the liquid. For example, for He one needs to go to much lower tem- peratures, below 2.6 mK, to find its superfluid state. This state is associated with the condensation of

He atom pairs, similar to the Cooper pairing of electrons in a superconducting state.[17] Because

He and

He are practically identical in all properties but statistics, a thorough comparison of these two substances is particularly interest- ing to understand the role quantum statistics plays in the behavior of many-body systems.

Helium can appear both as a bulk liquid and as droplets. At low temperatures, a mixed

He-

He drop is a structured cluster with

He in its core and

He in its outer shell.

One attractive feature of these drops is that they can be doped with impurities with relative ease for spectroscopic probing of the helium environment.[2–7] Depending on the impurity it will either go to the center of the drop, reside at its surface, or sink down to the

He–

He interface.[8] The structure of

He,

He and, to a lesser extend,

He–

He mixed droplets and liquids around atomic impurities or free electrons has been studied both in the experimental and theoretical fronts.

The dynamics of these systems, on the other hand, have proven elusive for many years. One may naively argue that, because few experiments are carried out to study the response of helium at the picosecond scale, an accurate dynamical description of helium systems is a purely academic issue of little relevance to the experimental community.

However, many experiments such as Prof. Maris’ observation of disappearing excited 1P electron bubbles[9] or Prof. Drabbels’ measurement of the desorption velocity of Ag atoms inside drops[10] have shown that a complete and accurate theoretical description of the dynamical processes underlying these experiments is fundamental to understand them.

The publications presented in this thesis are split in two chapters: Chapter 2 addresses some problems on the structure of drops and the mixed liquid. The differences between

3

He and

He are studied through the dipole absorption spectrum of Na and Mg impurities and through their effect on the moment of inertia of solvated carbonyl sulfide molecules.

Chapter 3 presents the development of an accurate and consistent framework to address time-dependent problems in helium liquids and clusters of some thousand atoms. This methodology is used to help interpret the experiments by Profs. Maris and Drabbels commented above.

1.1 Theoretical description of helium systems

A good theoretical description is essential to fully understand the experimental evidence.

Any potential description must take into account the key role of quantum mechanics and the strong He–He correlations.

There are several approaches to a theoretical exploration of the properties of many-

body systems. One possibility is to solve numerically the N -body Schr¨ odinger equation

by Quantum Monte Carlo simulation techniques.[18, 19] These techniques partially

alleviate the prohibitive scaling of computational cost with the number of particles that solving the Schr¨ odinger equation has. These are ab initio procedures that give a very detailed description of the groundstate of the system but require humongous amount of computational power to describe systems of experimental interest such as drops with some thousands of atoms. Other ab initio techniques include Green’s Function, Propagator Methods, Correlated Basis Function or Coupled Cluster Method.[20–22] These methods involve either diagramatic expansions or variations of a trial wavefunction, and are either less suited or just not suited for complex setups like a cluster doped with a strongly interacting impurity.

One alternative to these microscopic approaches is to pursue a phenomenological description of the system by using Density Functional Theory. In this approach, all the properties of the liquid are encoded in an energy functional which depends only on the one-body density instead of the complete N -body wavefunction.[23–26] This provides a good scaling with the number of particles while maintaining a wide applicability. The price to pay is the added complexity of finding a realistic energy functional and the limitation of having an hydrodynamic description of the system, with no information on the atoms themselves nor their correlations.

We have chosen the later approach in all the works presented here as its accuracy and efficiency have been proven. Besides, it is the only realistic way of computing real-time dynamics in the picosecond domain for a system of thousands interacting He atoms so far.

1.1.1 Density Functional Theory

Density functional theory is a rigorous formulation of nonrelativistic quantum many-body physics in which the energy of the system is taken as a functional of the one-body density and not the complete N -body wavefunction. The base for this formulation is the demonstration [27] by Hohenberg and Kohn in 1964 that the groundstate properties of a quantum system of interacting particles can be characterized completely by the one-body density through the appropriate energy functional.

Several energy functionals for a system of interacting helium atoms have been devel- oped over the years. The quintessential functional when dealing with inhomogeneous

4

He settings is a finite-range functional known as “Orsay-Trento” presented in 1995 by F. Dalfovo et al.[28] The extension to inhomogeneous

He–

He mixtures made in [29]

(which reduces to Orsay-Trento in the absence of

He) has been the functional of choice for the works presented in this thesis.

With an energy functional at hand, this theory can be easily extended to a Time-

Dependent Density Functional Theory (TDDFT). In its simplest implementation, the

dynamics of the system is obtained by minimizing the quantum action for the effective

macroscopic wavefunction Ψ, A[Ψ] =

dt



E[ρ] +



d r ¯ h

2m |∇Ψ|

− i¯hΨ

∂

∂t Ψ



,

where Ψ is a complex field whose squared modulus equals the one-body density, ρ = |Ψ|

. This minimization yields a time-dependent Schr¨ odinger-like equation for Ψ( r, t) of the form

i¯ h ∂

∂t Ψ( r, t) =



− ¯ h

2m ∇

+ δE δρ( r, t)



Ψ( r, t) .

which can be efficiently solved numerically by generic differential equation methods such as a predictor-corrector algorithm.[30]

In all the problems addressed in this work helium is coupled to an impurity, which may

be either an atomic impurity such as Ag or an excess electron trapped in a bubble. This

coupling is introduced as a helium-impurity potential also entering in the time evolution

equation for the impurity. The power and suitability of TDDFT for the dynamics of

liquid helium and droplets becomes apparent since the same formalism has allowed to

couple, with only small changes, the helium evolution with an adiabatically evolving

wavefunction,[14] a real-time evolving wavefunction,[15] o a moving classical particle with

spin-orbit degrees of freedom.[10]

Structure

2.1 Absorption spectrum of atomic impurities in isotopic mixtures of liquid helium

Resumen (Spanish)

El espectro de absorci´ on de impurezas at´ omicas en mezclas isot´ opicas de helio l´ıquido se describe en este trabajo mediante c´ alculos basados en un formalismo del funcional de la densidad a temperatura cero. Se consideran dos casos: en el primero, el espectro de absorci´ on de ´ atomos de Na ligados a gotas de

He

–

He

, con valores de N

de 100 a 3000 se presenta como caso de estudio de una impureza que no solvata en gotas de helio.

En el segundo, el espectro de absorci´ on de ´ atomos de Mg solvatados en mezclas isot´ opicas de

He y

He se presenta como caso de estudio de impureza disuelta en el l´ıquido uniforme.

Encontramos que el espectro de las impurezas se ve poco afectado por la composici´ on isot´ opica del entorno, y que depende b´ asicamente de la cantidad de helio que tiene a su alrededor sin importar el is´ otopo. En el caso de mezclas de l´ıquido uniforme, los resultados se presentan en funci´ on de la presi´ on para diferentes valores de concentraci´ on de

He. Los resultados para los l´ıquidos isot´ opicamente puros de

He y

He se comparan con los datos experimentales disponibles en la literatura.

5

Absorption spectrum of atomic impurities in isotopic mixtures of liquid helium

David Mateo, Alberto Hernando, Manuel Barranco, Ricardo Mayol, and Mart´ı Pi

Departament ECM, Facultat de F´ısica, and IN²UB, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain (Received 7 February 2011; published 6 May 2011)

We theoretically describe the absorption spectrum of atomic impurities in isotopic mixtures of liquid helium within a zero-temperature density functional approach. Two situations are considered. In the first one, the absorption spectrum of Na atoms attached to⁴He1000-³HeN3droplets with N³values from 100 to 3000 is presented as a case study of an impurity that does not dissolve into helium droplets. In the second one, the absorption spectrum of Mg atoms in liquid³He-⁴He mixtures is presented as a case study of an impurity dissolved into liquid helium. We have found that the absorption spectrum of the impurity is rather insensitive to the isotopic composition because the line shift is mostly affected by the total He density around the impurity, not by its actual composition. For bulk liquid mixtures, results are presented as a function of pressure at selected values of the³He concentration. The results for isotopically pure³He and⁴He liquids doped with Mg are compared with available experimental data.

DOI:10.1103/PhysRevB.83.174505 PACS number(s): 67.60.−g, 78.40.−q, 32.30.Jc

I. INTRODUCTION

The study of the absorption spectrum of impurities in liquid helium and its droplets has drawn considerable interest because it is a powerful tool to investigate the structure of the dopant-liquid complex, having become a classical field in optical spectroscopy. The optical properties of impurities in liquid⁴He and its droplets have been reviewed in Refs.1 and2, respectively.

Electronic spectroscopy studies have been carried out for atomic impurities in⁴He, and to a lesser extent, in³He.^3–5 Only very recently, the electronic absorption spectrum of an atomic impurity—a Ca atom—in mixed ³He-⁴He droplets has been reported and analyzed within density functional (DF) theory.⁶ A distinct feature of Ca atoms in mixed helium droplets is that, depending on the size and isotopic composition of the droplet, it may reside at the ³He-⁴He interface. Therefore, one would expect that its electronic spectrum might shed light on the structure of that interface, as this spectrum is affected by the liquid environment around the impurity.

Experiments on doped mixed droplets have to face the serious problem of determining the actual composition of the system. This is not easy because of the large number of atoms, mostly of³He, that are evaporated off the droplet after the dopant pick-up, altering the initial composition of the droplet in a way that is difficult to ascertain. The initial composition is not easy to determine either. In contrast, experiments in liquid mixtures may be carried out under well-controlled conditions, fixing, e.g., the³He concentration x3= N3/(N3+ N4) and particle density ρ = (N3+ N4)/V and temperature (T ) of the mixture, which in turn determine the total pressure (P ) throughout the equation of state of the fluid.

In this work, we aim to study the effect of isotopic composition on the absorption spectrum of atomic impurities in both finite (droplets) and extensive (liquid) helium systems.

We present results for the 3p ← 3s transition of Na attached to³He-⁴He droplets, complementing those we have previously published for Ca.⁶ It is well known experimentally⁷ and

theoretically⁸that, because of the limited solubility of³He in

4He at low temperatures,⁹mixed droplets have a core-shell structure made of nearly pure ⁴He and ³He, respectively.

Since Na atoms do not dissolve into helium droplets, the a priori most interesting situation is when the number of

3He atoms, N3, is rather small as compared to that of

4He, N4. Otherwise, the environment around Na is made of pure³He and one should not expect any difference with the absorption spectrum of Na in isotopically pure ³He droplets.³

At variance with the droplet situation,³He segregation in liquid helium mixtures at low temperatures only appears for concentrations above a critical value that depends on pressure.⁹ Hence, it is plausible that the absorption spectrum could be sensitive to the x3 value of the mixture. To check this hypothesis, we present calculations of the absorption spectrum around the 3s 3p¹P1← 3s^{2 1}S0transition of Mg atoms in liquid helium mixtures for selected values of x3and P .

This work is organized as follows. In Sec.II, we briefly recall the DF method used to obtain the structure of doped helium mixtures, drops, and bulk liquid as well, and the procedure used to determine the absorption spectrum incor- porating shape fluctuations of the liquid bubble around the dopant. The absorption spectrum of Na in ⁴He₁₀₀₀-³He_N3 droplets with N3= 100 to 3000, and that of Mg in ³He-

4He liquid mixtures for selected values of x3 and P , is discussed in Sec. III. Finally, a summary is presented in Sec.IV.

II. METHOD

A. Density functional description of the ground state of doped isotopic mixtures

The energy of the Na-droplet complex is written as a functional of the Na wave function (r), the⁴He effective macroscopic wave function (r) =√

ρ4(r), where ρ4(r) is the ⁴He atomic density normalized to N4 atoms, and the

3He particle and kinetic energy densities ρ3(r) (normalized to N3 atoms) and τ3(r).¹⁰ We have used a Thomas-Fermi

FIG. 1. (Color online) Three-dimensional view of Na@⁴He1000+³He_N3 droplets for different N3values. Also shown is the probability density of Na in arbitrary units.

approximation to write τ3(r) as a function of ρ3(r) and its gradient.³Within the pair potential approximation, we have¹¹ E[,,ρ3,τ3]= ¯h²

2 m⁴He



dr |∇(r )|² +



dr E(ρ4,ρ3,τ3)+ ¯h² 2 mNa



dr |∇(r )|² +



dr dr^|(r)|²VX²(|r − r^|) ρ3+4(r^).

(1) The Na-He X² pair potential has been taken from Ref.12.

The equations resulting from the variations of Eq. (1) with respect to , , and ρ3(r) are self-consistently solved as indicated in Ref.11.

Figure 1 displays a three-dimensional view of Na@⁴He₁₀₀₀+³He_N3 droplets for several N3values, and the probability density of Na,|(r )|², in arbitrary units. The figure shows the above-mentioned core-shell distribution of⁴He and

3He atoms in the droplet, as well as the known result that Na does not dissolve into them. In this respect, it is illustrative to compare the results for Na with those for Ca (see Fig.2 of Ref. 6corresponding to Ca@⁴He₁₀₀₀+³He₂₀₀₀). So far, Ca is the only known impurity that is dissolved into⁴He but not into³He droplets, and for this reason it may sink into the fermionic component until reaching the surface of the bosonic core. More attractive impurities like OCS reside in the bulk of the core, thus they are instrumental in the discussion of superfluidity at the nanoscale.⁷Here we discuss the remaining case of an impurity that resides at the surface of the droplet irrespective of the isotope. One of the cuts in Fig.1displays the pure³He-⁴He interface showing the building up of the

3He shell as N3increases. It is worth noting that, with a core of 1000⁴He atoms, a large amount of³He (N3∼ 2000) is needed before the density of the fermionic shell reaches that of liquid³He at saturation; see also Ref.8. The other cut displays the doped³He-⁴He interface. Notice that for the N4= 1000 droplet, one should not expect that the absorption spectrum of Na onto⁴He1000+³He_N3 differs much from that of Na onto an isotopically pure³He droplet of similar size if N3 1000.

The starting point for describing a Mg atom in liquid helium mixtures is also Eq. (1). The Mg-He X¹ pair potential has been taken from Ref.13. In this case, instead of fixing the number of atoms N3and N4, the asymptotic ρ3and ρ4densities far from the impurity have been fixed to those of the undoped mixture. In practice, we work at fixed P and x3values, which in turn fix the ρ3and ρ4values (and the corresponding chemical potentials) through the T = 0 equation of state supplied by our DF.¹⁰Details of the procedure and method used for solving the variational equations can be found in Ref.14for an electron bubble in liquid⁴He. The generalization to helium mixtures and atomic impurities is straightforward.

We want to emphasize that our method yields a self- consistent and accurate description of the thermodynamics of undoped liquid mixtures at zero temperature, a necessary starting point to address the properties of the doped system.

In particular, it reproduces the T = 0 phase diagram of the mixture. Figure2shows the calculated phase diagram obtained as explained in Ref.15.

The top and bottom panels of Fig.3show the helium density profile around Mg in the case of isotopically pure⁴He and

3He liquids for three different pressures. It can be seen that the helium density is strongly modulated around the impurity, slowly evolving toward the bulk liquid density as the distance

0 10 20 30 40

0 5 10 15 20

x3(%)

Segregation Line Spinodal Line

P (bar) Stable Metastable

FIG. 2. Calculated phase diagram of the⁴He-³He liquid mixture at T = 0. The solid line between stable and metastable regions is the maximum solubility line of³He into⁴He, and the dashed line is the spinodal line.

from the impurity increases. Notice also how the liquid density increases as P does, and how the radius of the bubble around the impurity is slightly larger for³He than for⁴He because of the surface tension being smaller for³He than for⁴He. This figure shows that, even for rather weakly interacting dopants such as Mg, the actual structure of the liquid cannot be easily guessed or represented by simple parametrizations. This is especially so in the case of isotopic liquid mixtures; see the middle panel of Fig.3.

These density profiles already give a first idea of what to expect from the study of the absorption peak as a function of P : the shift increases as the density does, and therefore it will also increase with P . Similarly, the shift should be larger for Mg in isotopically pure⁴He than in isotopically pure³He at given P , as the He-Mg interaction is the same irrespective of the isotope. These facts have been established experimentally.^4,5

The middle panel of Fig.3shows the density profile at P = 10 bar for x3= 9%. It can be seen from Fig.2that these conditions allow us to carry out the calculation for nearly the largest possible³He concentration before segregation. For the sake of comparison, the result for pure ⁴He is also shown.

Whereas the results displayed in the other panels of Fig.3 are known to some extent, to the best of our knowledge the density profiles of isotopic mixtures of liquid helium around an attractive impurity have not been previously determined.

Along the lines of the other two panels, one would expect a very weak dependence of the atomic shift on the composition of the mixture. Comparing the total helium densities displayed in the middle panel of Fig.3, the atomic shift might be slightly larger for liquid⁴He than for the mixture at the same pressure.

In the next section, we address these issues in detail, confirming these expectations.

We would like to close the discussion of the density profiles by pointing out two interesting characteristics of the liquid mixture at low temperatures, relevant for the forthcoming discussion of the absorption spectrum. The first feature is that, as can be seen from Fig.3, substituting ⁴He by³He atoms at a given P does not result in a sizable change in the liquid

0 0.01 0.02 0.03 0.04

r (˚A)

|φM g|²

0 0.01 0.02 0.03 0.04

P = 0 bar 10 bar 20 bar

4

He

0.01 0.02 0.03

ρ(˚A−3)

x3= 0%

3He at x3= 9%

4He at x3= 9%

x3= 9%

|φM g|²

3

He

P = 0 bar 10 bar 20 bar

0 5 10 15 20

|φM g|²

FIG. 3. (Color online) Selected density profiles of the liquid helium mixture around an Mg impurity whose probability density in represented in arbitrary units. Top panel: pure⁴He. Bottom panel:

pure³He. Middle panel: density profiles at P = 10 bar for x3= 9%.

Solid line, total density; thin dashed line,⁴He density; dotted line,

3He density. For the sake of comparison, the profile of isotopically pure⁴He is also shown (dashed line).

total density. This is due to the high incompressibility of liquid helium. The other feature worth noting is that³He atoms do not segregate around the impurity coating the surface of the bubble.

This is due to the large zero-point energy that a³He atom would have in such a small cavity, and it is more marked the larger the impurity-helium interaction is. As a consequence, the impurity bubble is coated by⁴He atoms and not by³He, in spite of the density increase of both isotopes at the first solvation shell, allowing the appearance of the rovibrational spectrum of OCS molecules in helium droplets⁷that otherwise would be quenched by the presence of normal-phase³He atoms. This is at variance with the situation at the free surface of a mixed drop or liquid mixture, where the existence of Andreev states

bearing a large capacity of hosting³He atoms makes possible the accumulation of this isotope at the surface.⁸

B. Absorption spectrum

To determine the absorption spectrum of an impurity atom embedded in a condensed system, it is customary to use Lax’s method,¹⁶ together with the diatomics-in-molecules approach.¹⁷This is basically the method we have followed,¹¹ once the ground state (gs) of the dopant-helium mixture (droplet or bulk liquid) has been determined. The ² and

2 excited pair potentials for Na-He are from Ref.18, and the

1 and¹ ones for Mg-He are from Ref.19. In the case of Na, we have also considered the spin-orbit splitting.¹¹

With only these ingredients, the model yields a good description of the absorption energies—provided the pair potentials are accurate enough—but the shape, especially the width, of the absorption line is poorly reproduced.

The well-known reason for this drawback is the neglect of the coupling of the impurity dipole excitation to the shape fluctuations (modes) of the liquid cavity around it.

Including this coupling in the calculation yields a much better agreement with experiments. This is illustrated, e.g., in Ref. 20 for Mg atoms in ⁴He droplets and in Ref. 21 for electron bubbles in liquid⁴He. Taking into account shape fluctuations is very cumbersome if the impurity bubble is not spherical. The situation is far more complex for liquid³He and mixtures because the modes of the cavity are difficult to determine.

Shape fluctuations are effortlessly calculated in quantum Monte Carlo simulations of the absorption spectrum^22–24 by taking advantage of the information carried out by the quantum “walkers.” Somewhat inspired by this atomiclike simulation, an easy-to-implement method has been proposed within DF theory to include shape fluctuations, and it has been applied to the case of Cs in liquid⁴He,²⁵and was later adapted to the droplet geometry.^26,27The extension to the case of isotopic mixtures is straightforward, but for the sake of completeness we present it here as applied to the case of a Na impurity, outlining the method we have followed to determine the absorption spectrum of an impurity in liquid helium.

The Born-Oppenheimer approximation allows the factor- ization of the electronic and nuclear wave functions, and the Franck-Condon approximation allows the positions of the atomic nuclei to remain frozen during the electronic transition.

Within these approximations, the line shape for an electronic transition from the gs to an excited state (ex) is obtained as the Fourier transform of the time-correlation function,

I (ω) ∝

m



dte^{−i(ω+ωgs)t}



d³r^gs∗e^(it/¯h)Hmex^gs, (2) where ¯hω^gsand ^gs(r) are the eigenenergy and eigenfunction of Na in its gs, respectively. The Hamiltonian is H_m^ex= Tkin+ V_m^ex(r), where Tkinis the kinetic energy operator and V_m^ex(r) is the potential energy surface defined by the mth eigenvalue of the excited potential matrix V (r) = U (r) + VSO, where U (r) is the convolution of the excited pair potentials ² and² with the total helium density ρ(r), as the³He- and

4He-impurity pair potentials are the same, and VSOaccounts for

the spin-orbit coupling.¹¹Introducing ^gs(r)=

νa_ν^m^m_ν(r) in Eq. (2), where ^m_ν(r) are the eigenfunctions of H_m^ex and a^m_ν =

d³r ^m_ν(r)^∗^gs(r) are the Franck-Condon factors, we obtain

I (ω) ∝

m



dte^{−i(ω+ωgs)t}

ν

a^m_ν²e^iωmνt

=

m



ν

a_ν^m²δ

ω + ω^gs− ω_ν^m

, (3)

where ¯hω^m_ν are the eigenvalues of H_m^ex.

If the Franck-Condon factors arise from the overlap between the gs and excited states with large quantum numbers, corresponding to the continuous or quasicontinuous spectrum of H_m^ex, we can assume that Tkin  V_m^ex, and the Hamil- tonian is approximated by H_m^ex∼ V_m^ex(r). Introducing this approximation in Eq. (2) and integrating over time, we get the semiclassical expression for I (ω),

I (ω) ∝

m



d³r|^gs(r)|²δ(ω − [V_m^ex(r)/¯h − ω^gs]) . (4) We have evaluated this expression as follows. First, the helium distribution is stochastically represented by a large number of configurations nc, of the order of 10⁶. Each configuration consists of a set of N positions for the He atoms in the sampling box and one for the impurity. These positions are randomly generated by importance sampling techniques, using the DF helium density ρ(r)/N as the probability density distribution, plus a hard-sphere repulsion between He atoms to approximately take into account He-He correlations. The diameter of the sphere has to be of the order of h = 2.18 ˚A to be consistent with the DF description of the liquid, as h is the length used in the functional to screen the Lennard-Jones interaction between particles and to compute the coarse- grained density.¹⁰We have chosen a density-dependent sphere radius of the form

Ri= R(ri)= h 2

ρ0

ρ(r¯ i) _1/3

, (5)

where ρ0is the saturation density value and ¯ρ is the coarse- grained density, defined as the averaged density over a sphere of radius h. Although this scaling has no effect in the bulk, it is fundamental to correctly reproducing the density in the droplet surface region. The rationale for choosing this Riis sketched in the Appendix. Lastly, the position of the impurity is also randomly generated using|^gs(r)|²as the probability density distribution.

To determine the line shape, we obtain for each configu- ration{j} the V_m^ex{j} eigenvalues of the excited-state energy matrix 

iU (|r^{j}_i − r^{j}_Na|) + VSO [Eq. (16) of Ref.11] and subtract from them the pairwise sum of the gs pair potential interactions V^gs{j} =

iVX²(|r^{j}_i − r^{j}_Na|) to obtain the excitation energy. The histogram of the collected stochastic energies is identified with the absorption spectrum, i.e.,

I (ω) ∝

m

1 nc

nc



{j}

δ ω −

V_m^ex{j} − V^gs{j} /¯h

. (6) In this way, we obtain the absorption spectrum of impurities in liquid helium including shape fluctuations. When this is the

main source of broadening, as for impurities embedded in the liquid or in the bulk of drops, the method has proved to repro- duce fairly well the broadening of the absorption line, as we show for Mg in Sec.III. Note that other sources of broadening such as thermal wandering²⁰or droplet size distribution effects may have a sizable influence for impurities residing in the outer surface of the droplet, and they are not accounted for by this procedure.

III. ABSORPTION SPECTRUM RESULTS A. Na in mixed helium droplets

Figure 4 shows the absorption spectrum for Na@⁴He1000+³He_N3 mixed droplets with N3= 100,500,1000, and 3000. The vertical lines represent the location of the absorption lines of the free Na atom. As expected, the shift in the spectrum increases with the number of³He atoms.³In the⁴He₁₀₀₀+³He₁₀₀droplet, the effect of

3He is barely perceptible and its spectrum is sensibly that of Na in the isotopically pure⁴He droplet. One might expect the impurity to draw the³He atoms and be quickly surrounded by them, but this is not quite so even for a more attractive impurity such as Ca.^24,28

In the N3= 1000 and 3000 drops, it is the ⁴He core that plays no significant role, and the spectrum is sensibly that of the isotopically pure ³He droplet.²⁹ The N3= 500 droplet is an intermediate case, in which there is enough

3He to influence the absorption spectrum but the number of

3He atoms is still small, and the density in the ³He shell does not reach that of the liquid at saturation; see, e.g., Ref.8and Fig.1. In this configuration, the shift is slightly smaller than in a pure³He drop, although the difference is too small to be detectable. We recall that the experiments have been carried out for isotopically pure droplets of about 5000 atoms. The calculated peaks are narrower than in the experiment because Na resides at the outer surface of the droplet and thermal wandering and droplet-size distribution

0.01 0.02 0.03

16950 16960 16970 16980

Intensity

ω (cm⁻¹)

N3= 100 N³= 500 N3=1000 N3=3000

N⁴=1000

FIG. 4. (Color online) Absorption spectrum (arbitrary units) of Na in ⁴He1000+³He_N3 droplets with N³= 100 (solid line), 500 (dashed line), 1000 (dotted line), and 3000 (dash-dotted line). The thin vertical lines represent the gas-phase transitions. The spectra are normalized so that the more intense peaks all have the same height.

0.05 0.1 0.15

In tensit y

0.05 0.1 0.15

35000 35500 36000 36500

ω (cm

)

This Work Ref. 30 Ref. 4

4

He

3

He

Total

FIG. 5. (Color online) Top panel: Calculated absorption spectrum (arbitrary units) of Mg in liquid ⁴He at P = 0 bar (solid line) compared to the experimental results of Refs.30(dashed line) and5 (dotted line). The spectra are normalized so that the peaks have the same height. Bottom panel: Calculated absorption spectrum (arbitrary units) of Mg in liquid³He at P = 0 bar. The line has been decomposed into its two  components and one  component, the latter one being the higher-energy transition. The thin vertical line represents the gas-phase transition.

effects should contribute to the broadening in a non-negligible way.¹¹

B. Mg in liquid helium mixtures

Isotopic liquid helium mixtures are better suited than mixed drops to determine the effect of the isotopic composition on the absorption line, as one avoids finite-size effects and the actual composition of the fluid sample can be controlled. In addition, they offer the possibility to study the pressure effect on the spectrum.

The reason for choosing Mg atoms for this study is twofold.

First, there are detailed results for its absorption spectrum in pressurized isotopically pure liquid³He and⁴He,^5,30indicating that the shift in the absorption peak is about 100 cm⁻¹smaller in ³He than in ⁴He.⁵ It is thus reasonable to expect that the absorption spectrum may show some sensitivity to the isotopic composition of the mixture. Secondly, the adiabatic Mg-He pair potentials for the ground¹³ and excited states¹⁹ are known with good accuracy. We want to mention the existence of a series of recent studies of Mg in helium droplets aimed at ascertaining whether this impurity resides in the

bulk or at the surface of ⁴He drops.19,20,31–33 Most studies point toward a sizable radial delocalization of Mg inside large drops.

In a first stage, we have computed the absorption spectrum of Mg in isotopically pure liquid⁴He and³He. For the former, there are two inconsistent sets of experimental data obtained by the same group, both of which are compared with our calculations in the top panel of Fig.5. No detailed results for the line shape in the case of³He have been published for comparison. We remind the reader that our calculations are at T = 0, whereas the experiments have been carried out at 1.4 K.

While our calculations compare very well with the ex- perimental results for ⁴He in Ref.30, they are blueshifted with respect to those of Ref. 5 for ⁴He and ³He as well.

Despite this discrepancy, we have found, in agreement with the experimental findings,⁵ that the shift is 0.77 nm larger in bulk ⁴He than in bulk ³He. This is an important check to assure that the calculation may disclose effects associated with the isotopic composition of the liquid mixture, as shown below.

The pressure dependence of the absorption spectrum of Mg in isotopically pure liquid ⁴He and ³He is shown in Fig. 6 for P = 0, 10, and 20 bar, and the peak energy is represented in Fig.7 as a function of pressure. This dependence is in

0 0.2 0.4 0.6 0.8

0 0.2 0.4 0.6 0.8

35000 35500 36000 36500

Intensity

ω (cm⁻¹)

P = 0 bar 10 bar 20 bar 4

He

3

He

FIG. 6. (Color online) Top panel: Calculated absorption spectrum (arbitrary units) of Mg in liquid⁴He at P = 0 (solid line), 10 (dashed line), and 20 bars (dotted line). Bottom panel: Same as top panel for

3He. The thin vertical line represents the gas-phase transition. The spectra are normalized so that the peaks have the same height.

300 400 500 600

0 5 10 15 20

Shift(cm−1)

P (bar)

FIG. 7. (Color online) Peak energy shift of the absorption spectrum of Mg in isotopically pure liquid⁴He (squares) and³He (dots) as a function of pressure. The crosses represent the peak energy shift slightly below the segregation line for the corresponding pressures; the x3values are 6.3%, 8.9%, 9.4%, 9.2%, and 8.4% for P = 0, 5, 10, 15, and 20 bar, respectively. Least-squares linear fits for each set of points are drawn as a guide to the eye.

qualitative agreement with experiment,⁵although our results are systematically blueshifted with respect to the experimental results by about 1 nm. The crosses in Fig.7represent the peak energy slightly below the zero-temperature segregation line as a function of pressure. Thus, each point corresponds to a different x3value, as shown in Fig.2. We conclude that the shift in the peak energy of the Mg absorption spectrum is significant. However, the size of the line can make it hard to determine experimentally the x3dependence. This is illustrated in Fig.8, where we have drawn the absorption spectrum of Mg in an x3= 9.4% mixture at P = 10 bar. The absorption peak is shifted by 28 cm⁻¹ with respect to the isotopically

0.2

35000 35500 36000 36500

Intensity

ω (cm⁻¹) x3= 9.4%

x3= 0%

P = 10 bar

0.1

FIG. 8. (Color online) Absorption spectrum (arbitrary units) of Mg in an x³= 9.4% liquid mixture at P = 10 bar (solid line). The dashed line is the result for the isotopically pure liquid⁴He at the same pressure. The thin vertical line represents the gas-phase transition.

The spectra are normalized so that the peaks have the same height.

pure liquid⁴He, in spite of the small morphological changes in the density profile introduced by this small ³He amount (see Fig.3).

IV. SUMMARY

We have studied the absorption spectrum of atomic impuri- ties in mixed helium drops and liquid helium mixtures, paying special attention to the dependence of the spectrum on the³He concentration. For the study of drops, we have chosen Na as the impurity, which always resides on the surface of the drop, complementing the studies carried out in Ref.6for Ca.

For droplets of the size and composition addressed herein, we have found that the shift in mixed droplets is larger than in⁴He droplets but slightly smaller than in³He droplets. Even though a large amount of³He is needed for the density in the outer shell of the mixed droplet to reach the bulk liquid³He value, our results indicate that the spectrum of the impurity is very insensitive to the isotopic composition and it rapidly saturates to the value of pure³He droplets when the quantity of

3He is increased. From this we infer that the effect of isotopic composition on the absorption spectrum is hardly detectable for alkali impurities in helium droplets. We have chosen Mg for the study of liquid mixtures, and we have compared the results obtained for isotopically pure liquid⁴He and³He with the experiments reported in Refs.5and30. We have found that the peak energy in saturated helium mixtures can be shifted by up to some tens of cm⁻¹ from that in pure ⁴He at the same pressure. While much smaller atomic shifts from the gas- phase value have been detected experimentally, to determine the dependence of the atomic shift on the isotopic composition of the mixture is an experimental challenge due to the large width of the absorption line.

Finally, we would like to point out that the infrared spectrum of excess electrons might be a way to determine the structure of electron bubbles in isotopic mixtures of liquid helium, as it has been for isotopically pure liquid⁴He or³He.³⁴ Due to the electron-helium repulsion, electron bubbles are fairly large, with a radius of about 18.5 ˚A for⁴He and 22.5 ˚A for

3He.³⁵ At variance with the situation for the small bubbles around an atomic impurity in liquid helium mixtures, the electron bubble surface should be coated by ³He, as it is for bubbles appearing in homogeneous cavitation processes.¹⁵ This coating increases the bubble radius with respect to that of isotopically pure⁴He, as it decreases the surface tension of the liquid. Since the electron spectrum is very sensitive to the bubble radius, determining it would probe the structure of the electron bubble in the mixture. Knowledge of this

structure has potential implications for cavitation in liquid helium mixtures.^36,37Work is in progress to obtain the electron absorption energies in liquid helium mixtures.

ACKNOWLEDGMENTS

We would like to thank Frank Stienkemeier and Oliver B¨unermann for useful exchanges. This work has been per- formed under Grants No. FIS2008-00421/FIS from DGI, Spain (FEDER), and No. 2009SGR01289 from Generalitat de Catalunya. D.M. has been supported by the ME (Spain) FPU program, Grant No. AP2008-04343. A.H. has been supported by the MICINN (Spain) FPI program, Grant No. BES-2009-027139.

APPENDIX

Ignoring normalization, which is irrelevant for the present discussion, the probability distribution we have chosen for sampling N helium atoms is

P^N({ri}) = N i=1

ρ(ri) N

N j <i

(rij− h), (A1) where  is the step function. The atomic density 

iδ(r − ri) corresponding to this probability distribution is _N



i=1

δ(r − ri)



= ρ(r)

 ⎛⎝^N−1

j =1

drj(|rj− r| − h)

⎞

⎠ P^N−1({ri}) = ρ(r).

(A2) Hence, due to the He-He correlations introduced in P^N, the density of the system is not equal—and cannot be—to the DF particle density ρ(r), and one has to do something to recover ρ(r) back from the sampling. To do so, we have introduced a density dependence on h such that the integral appearing in Eq. (A2) is a constant that could be absorbed in the normaliza- tion. This cannot be done exactly, but if one assumes that ρ(r) varies smoothly, i.e.,∇ρ(r)  ρ(r)/h, then the result of the integral can be written as a power series of h³ρ(r), where ¯¯ ρ(r) is the coarse-grained density. Then, to turn the integral into a constant, we just need to add a density dependence in h of the form h ∝ ¯ρ(r)^−1/3. This is the reason why we have chosen the hard-sphere radius R as expressed in Eq. (5).

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